#404824

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(25,26,28,32)(27,34,35,30)(29,31,37,33)(36,39,40,38);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,35)(29,36)(31,38)(32,34)(33,39)(37,40);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,21)(22,24)(27,35)(30,34)(36,40)(38,39);(17,19,20,23)(18,22,21,24)(26,31)(29,37)(30,38)(32,33)(34,39)(36,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,c^2=efi,d^2=gi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312020);